Directory UMM :Data Elmu:jurnal:I:International Journal of Production Economics:Vol68.Issue1.Oct2000:
Int. J. Production Economics 68 (2000) 1}14
Heuristic methods for cost-oriented assembly
line balancing: A survey
Matthias Amen*
Institut fu( r Unternehmensrechnung und Controlling, University of Berne, Engehaldenstrasse 4, CH-3012 Bern, Switzerland
Received 17 March 1998; accepted 21 September 1998
Abstract
This paper is concerned with cost-oriented assembly line balancing. This problem occurs especially in the "nal
assembly of automotives, consumer durables or personal computers, where production is still very labour-intensive, and
where the wage rates depend on the requirements and quali"cations to ful"l the work. First a short problem description is
presented. After that a classi"cation of existent and new heuristic methods for solving this problem is given. The heuristic
methods presented in this paper are described in detail. A new priority rule called `best change of idle costa is proposed.
This priority rule di!ers from the existent priority rules because it is the only one which considers that production cost
are the result of both, production time and cost rates. Furthermore a new sophisticated method called `exact solution of
sliding problem windowsa is presented. The solution process is illustrated by an example, showing how this metaheuristic
works together with an exact method. ( 2000 Elsevier Science B.V. All rights reserved.
Keywords: Assembly line balancing; Cost-oriented production planning; Heuristic methods
1. Introduction
In an assembly line the product units move with
a constant transportation speed through the consecutive stations. The total work content to be
performed by the production system has been split
up into economical indivisible work elements
which are called tasks. Among the set of tasks there
exist technological precedence relations. The set of
tasks to be performed in the same station is called
an operation or a station load. The time to perform
an operation is restricted by the cycle time. The
* Tel.: #41-31-631-85-01; fax: #41-31-631-37-80.
E-mail address: matthias.amen@iuc.unibe.ch (M. Amen)
assembly line balancing problem consists in allocating the tasks to the stations subject to the
technological precedence relations, the cycle time
restriction of the stations and the indivisibility of
the tasks.
In literature the objective usually is to minimize
the number of stations in a line for a "xed cycle
time [1, p. 650]. In other words: The objective is to
minimize the total idle time of the total capacity
provided by the sum of the stations of the line [2,
p. 911]. Therefore, this is called time-oriented assembly line balancing. The objective in cost-oriented
assembly line balancing which is considered in this
paper is to minimize the total cost per product unit
[3}6].
In general, assembly is a labour-intensive kind of
production. Therefore, in cost-oriented assembly
0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 9 9 ) 0 0 0 9 5 - X
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M. Amen / Int. J. Production Economics 68 (2000) 1}14
line balancing, the labour cost has to be analyzed in
detail: The wage rate of each operation depends on
the maximal point value of the tasks assigned to
a station. As the tasks di!er in their di$culty,
there are di!erences in the point values and
thus in the corresponding wage rates. The worker
has to be paid for the whole cycle time irrespective
of the duration of the operation. So the total
labour cost per product unit is the sum of the
wage rates of all stations multiplied by the cycle
time. Further it is assumed that the cost of
capital depends on the overall line length and that
all stations have the same length. All other costs
(e.g. material cost) can be assumed to be independent of the division of labour and of the line length
[3, p. 82]. Introducing the following symbols:
I
number of tasks (!)
i
index for the tasks, i"1(1)I
M number of stations (!)
m
index for the stations, m"1(1)M
set of tasks, assigned to station m, m"1(1)M
I4
m
c
cycle time (TU/PU)
k
total cost per unit (MU/PU)
cost rate of station m, m"1(1)M (MU/TU)
k4
m
k43 cost of capital per station (MU/PU)
k48 wage rate of station m, m"1(1)M (MU/TU)
m
cost rate of task i, i"1(1)I (MU/TU)
k5
i
k58 wage rate of task i, i"1(1)I (MU/TU)
i
(Note: Dimensions: TU, time units; PU, product
units; MU, monetary units; Labels: r, cost of capital; s, station; t, task; w, wage rate.)
The total cost per product unit can be calculated as
A
B
M
k" + ck48 #Mk43
m
m/1
with k48"maxMk58Di3Is N.
m
i
m
Since the conveyor speed is "xed by the cycle
time, the cost of capital for a single station can
be transformed into a cost rate per time unit.
The total cost of capital can be formulated as
Mk43"cM(k43/c). Thus for the purpose of capacity
balancing the relevant cost rates per time unit are
made up by the sum of the capital cost per time unit
and the task-depended wage rates: k5"k58#k4#/c
i
i
and k4 "k48#k4#/c. This simpli"es the calculation
m
m
of the total cost per product unit to
M
M
k" + ck4 " + c maxMk5Di3I4 N.
m
i
m
m/1
m/1
The total cost per product unit k"+M ck4
m/1 m
di!er from the term +I d5k5 only by the idle cost
i/1 i i
per product unit which are caused by idle time
and/or cost rate di!erences of the tasks assigned to
the same station. Therefore, the objective to minimize the total cost per product unit is equivalent to
minimize the total idle cost per product unit.
2. Classi5cation of heuristic methods for solving the
cost-oriented assembly line balancing problem
As the NP-complete bin-packing problem can be
reduced to the time-oriented assembly line balancing problem in polynomial time, the time-oriented
assembly line balancing problem is NP-hard [7,
p. 56]. This also holds for the cost-oriented problem [4, p. 479]. Thus it is justi"ed to develop
heuristic solution methods.
There exist two di!erent general approaches for
assembly line balancing. In the station-oriented approach in each step of the solution process only one
station is considered. Therefore we have to look for
tasks which have to be assigned to the current
station. In the task-oriented approach in each step of
the solution process a selected task is considered.
Therefore we have to look for the station to which
the current task has to be assigned to [8, pp.
917}918; 9, pp. 182}184]. All of the methods described in this paper use the station-oriented approach. Fig. 1 gives an overview of the existent and
new methods which can be classi"ed as follows:
Most of the methods can be characterized as
random choice/priority rule methods. In each step of
the solution process all of the methods of this
category but one choose one of alternative tasks for
assignment. For determining the task either random choice or priority rules are used. The priority
rule methods can be distinguished into methods
that make use of one problem-oriented priority
rule and methods that use several problem-oriented
priority rules. Priority rules are called problemoriented if they consider structural aspects of the
M. Amen / Int. J. Production Economics 68 (2000) 1}14
3
Fig. 1. Heuristic methods for cost-oriented assembly line balancing.
problem. The set of problem-oriented priority rules
consists of three subclasses. The time-oriented priority rules consider only the durations and/or the
precedence relations of the tasks, the cost rateoriented priority rules consider only the cost rates of
the tasks. Due to the fact that cost are the product
of both, cost-rate and duration, the new costoriented priority rule **best change of idle cost++
which will be presented in the next section takes
both aspects into account. In methods with several
problem-oriented priority rules the single rules are
used in lexicographic order, i.e. problem-oriented
priority rules are used as tie-breakers. All priority
rule methods use a simple non-problem-oriented
priority rule as a "nal tie-breaker.
One priority method di!ers from the other as in
the major steps of the solution process one of several alternative station loads has to be chosen by
the use of di!erent priority rules which consider the
situation in the alternative station loads. The alternative station loads are generated by the use of
several problem-oriented priority rules to choose
one of alternative tasks. Thus this method is a
further development of the previous mentioned
methods.
The new method called **exact solution of sliding
problem windows++ is quite di!erent from the heuristic methods mentioned before because it is a heuristic version of an exact method. With respect to the
similarities of the heuristic methods, the methods
are grouped into the following classes:
f Z methods with random choice task assignment;
f P methods with one problem-oriented priority
rule;
f H methods with several problem-oriented priority rules;
f F methods with exact solution of sliding problem
windows;
f E exact method.
Class H contains both, the simple methods which
choose one of alternative tasks and the more advanced method which makes use of these simple
methods when generating alternative station loads.
In the next section the methods of classes Z and
P and the simple methods of class H are presented.
Section 4 is concerned with the more advanced
method of class H. Section 5 deals with the methods
of class F. Class E consists of only one method. As
exact methods are beyond the topic of this paper
4
M. Amen / Int. J. Production Economics 68 (2000) 1}14
assignable tasks is empty, the current station
is **maximally loaded++ [10, p. 266; 8, p. 918].
Then a new station is established and the set of
assignable tasks has to be updated again. Each
method stops after all tasks have been assigned to a
station.
The random choice method (class Z) [11,
pp. 25}35 and 44}46] can be used several times for
solving the same problem instance. Among the set
of randomly generated solutions for the same problem instance only the best one has to be considered
for realization.
For a formal description of the priority rule
methods, the following additional symbols are introduced:
Fig. 2. Procedure for assigning tasks to stations.
this method is not presented in this paper (for
further reading see [5,6]. The exact method works
as a subalgorithm for the methods of class F.
I!44*'/ set of assignable tasks
duration of task i, i"1(1)I (TU/PU)
d5
i
F
set of all technological immediate following
i
tasks of task i, i"1(1)I
*k
change of idle cost if task i will be assigned
i
to the currently considered station, i3I!44*'/
(MU/PU)
r
ranked positional weight of task i, i"1(1)I
i
(TU/PU)
With this symbols the following problem-oriented
priority rules of class P can be de"ned:
z P-MaxD
3. Methods with choice of one of alternative tasks
All methods with choice of one of alternative
tasks in each step of the solution process follow the
same procedure when assigning tasks to stations.
The procedure is shown in Fig. 2.
As the station loads are generated successively, in
each step of the procedure only one station is considered (station-oriented approach). The tasks which
have no predecessor task or only predecessor tasks
already assigned to the current or to an earlier
station are called available. The available tasks
which have a duration not longer than the remaining idle time of the considered station are called
assignable. Among the set of assignable tasks one
task is chosen according to the method used. After
assigning this task the sets of available and assignable tasks are updated immediately. If the set of
z P-MaxR
Maximal task duration [12, p. 728],
i.e. maxMd5Di3I!44*'/N.
i
Maximal ranked positional weight
([13, p. 395 in the modi"ed version
of Hahn], [14, pp. 44}46]), i.e.
maxMr Di3I!44*'/N, where
i
G
d5#+ i r ,
j|F j
r" i
i
d5,
i
z P-MaxF
if F OH,
i
if F "H.
i
Maximal number of immediate
followers [12, p. 728], i.e.
max MDF D D i3I!44*'/N.
i
z P-MaxKt Maximal cost rate [4, p. 481], i.e.
maxMk5Di3I!44*'/N.
i
z P-MinKt Minimal cost rate [3, pp. 106}107],
i.e. min Mk5Di3I!44*'/N.
i
z P-MinKts Minimal absolute di!erence to the
current station cost rate [3, pp.
106}107], i.e. minMDk5!k4 D D i3I!44*'/N.
i
m
M. Amen / Int. J. Production Economics 68 (2000) 1}14
5
both, the cost rates and the time needed by the
workers of the line.
Within class H there exist three priority rule
methods with the choice of one of alternative tasks
in each step of the procedure. The lexicographic
orders of priority rules of these heuristic methods
are given below. For the heuristic methods of Steffen and Heizmann the lexicographic order di!ers
according to the fact whether the station is just
established (empty station) or the station has been
established at an earlier step in the solution process
(partially "lled station).
z H-Stef
Fig. 3. Calculation of *k .
i
z P-MinKI
Best change of idle cost, i.e.
minM*k Di3I!44*'/N, where
i
if k5)k4 ,
!d5k5 ,
m
i
i i
*k "
i
(k5!k4 )c!d5 k5 , if k5'k4 .
m
i
i i
m
i
The non-problem-oriented priority rule
G
z P-MinI
Minimal task number [12, p. 729],
i.e. min MiDi3I!44*'/N (Usually the
tasks are numbered topologically according to the precedence relations.)
is used as "nal tie breaker in all priority rule
methods. While applying the new priority rule `best
change of idle costa (P-MinKI) both decreases and
increases can occur, which has to be accepted. Not
choosing an assignable task with *k '0 and esi
tablishing a new station in some cases may cause an
extra station and therefore higher overall idle cost.
The calculation of *k is illustrated in Fig. 3.
i
P-MinKts and P-MinKI are dynamic rules because they depend on the current cost rate of the
station. For these rules it is necessary to calculate
not only the remaining idle time of the current
station after each assignment, but also to update
the cost rate of the station immediately. While
P-MaxD, P-MaxR and P-MaxF are only timeoriented and P-MaxKt, P-MinKt and P-MinKts
depend only on the cost rates, P-MinKI is the only
priority rule which considers that cost results from
Heuristic method of Ste!en [3, pp.
105}110]:
Lexicographic
Lexicographic order
order of priority of priority rules in
rules in case of case of already
just established established stations
stations (station (station is partially
is empty):
"lled with tasks):
1. P-MaxR
1. P-MinKts
2. P-MinI
2. P-MinKt
3. P-MaxR
4. P-MinI
z H-Heiz Heuristic method of Heizmann [15,
pp. 110}124]:
Lexicographic order
Lexicographic
order of priority of priority rules in
rules in case of case of already
just established established stations
stations (station (station is partially
"lled with tasks):
is empty):
1. P-MaxD
1. Identity of task
cost rate and current
station cost rate (i.e.
Dk5!k4 D"0 is
m
i
required).
2. P-MinI
2. P-MaxD
3. P-MinI
If there is no task which has the same
cost rate as the current station cost rate
in the case of an already established
station (station is partially "lled with
tasks), then the priority rule P-MaxD is
used together with the tie-breaking rule
P-MinI.
6
z H-WR
M. Amen / Int. J. Production Economics 68 (2000) 1}14
Heuristic method `wage ratea of Rosenberg/Ziegler [4, p. 481]:
1. P-MaxKt
2. P-MaxD
3. P-MinI
This heuristic method is a modi"cation
of the well-known `"rst-"t decreasing
heighta-method suggested for solving
two-dimensional packing problems [16,
p. 810]. The method is called `wage
ratea because Rosenberg/Ziegler [4]
consider only labour cost.
I!44*'/,105
m,w,u
d4,105
m,w,u
k4,105
m,w,u
"xed
4. A method with choice of one of alternative
station loads
A priority-rule method with choice of one of
alternative station loads is the `wage rate smoothinga method (H-WRS) [4, pp. 481}484]. The
method consists of a three-phase algorithm. In
order to give a formal description of this method
the following symbols are introduced:
I3%45
set of not yet "nally assigned tasks
I4,105,1)!4%2 set of tasks which de"nes the potential
m
station load of station m in phase 2,
m"1(1)M
I4,105,1)!4%3 set of tasks which de"nes the potential
m
station load of station m in phase 3,
m"1(1)M
=
number of di!erent cost rates (!)
w
index of the cost rate, w3M1,2, =N
set of assignable tasks with the cost
I!44*'/,l
w
rate kl , w"1(1)=
w
;
number of potential station loads for
w
the currently considered task cost rate
kl (!), w"1(1)=
w
u
index of the potential station loads,
u3M1,2, ; N
w
set of tasks which de"nes the uth poI4,,,105
m,w,u
tential station load for the considered
cost rate kl of the station m in the
w
phases 1 and 3, u"1(1); , w"1(1)=,
w
m"1(1)M
set of tasks which de"nes the uth potenI4,,,105,2
m,w,u
tial station load for the considered cost
rate kl of the station m in the phase 2,
w
u"1(1); , w"1(1)=, m"1(1)M
w
set of tasks with task cost rate kl that
w
are assignable to station m if
the current station load is de"ned
by the set of tasks I4,,,105, u"1(1); ,
m,w,u
w
w"1(1)=, m"1(1)M
work content of the current station
load I4,,,105 (TU/PU), u"1(1); ,
m,w,u
w
w"1(1)=, m"1(1)M
cost rate of the potential station
load I4,,,105 (MU/TU), u"1(1); ,
m,w,u
w
w"1(1)=, m"1(1)M
variable which indicates in phase 1
whether phase 1 has to be restarted or
whether phase 2 has to be started (!)
Fig. 4 shows the major steps of the 3-phase
algorithm. The formal descriptions of the phases
1}3 are given in the appendix.
In phase 1 of the procedure, potential station
loads are established. In phases 2 and 3 the potential station loads which have been calculated in
phase 1 are taken and further tasks are assigned. In
phase 1 each potential station load consists only of
tasks which have identical cost rates. The phases 2
and 3 are completely separated from each other. In
phase 2 a potential station load taken from phase 1
is "lled further only with tasks which have the same
cost rate as the station cost rate. In phase 3 a potential station load taken from phase 1 is "lled further
with tasks which could have cost rates di!erent
from the station cost rate of phase 1. In step 5 a station load has to be chosen from the two alternative
station loads which have been obtained from the
phases 2 and 3. After a potential station load is
"nally "xed, in step 5 the algorithm moves back to
phase 1. Within phase 1 stopping-criteria are to be
checked at two di!erent steps.
5. A method with exact solution of sliding
problem windows
A new heuristic method which is quite di!erent
from those described in the foregoing sections is the
`exact solution of sliding problem windowsa (class
F). This heuristic is an application of the `working
M. Amen / Int. J. Production Economics 68 (2000) 1}14
7
Fig. 4. Wage rate smoothing.
forward techniquea [17, pp. 315}316]. The key concept is to solve consecutive small problems by the
use of an exact method. A part of the optimal
solution of a small problem will be "xed "nally.
Then the next small problem will be de"ned and
solved optimally. This process continues until the
last small problem is solved. Similar methods for
solving di!erent assembly line balancing problems
were suggested in [18}20].
The idea to develop a heuristic application of an
exact method derives from the fact that all existent
heuristic methods load the stations maximally, but
that the cost-oriented optimal solution could be missed if the stations are "lled in this way [5, p. 225; 6,
1998a]. The exact method used is the only existent
method in which the stations are not necessarily
loaded maximally.
5.1. Formal description of the method
For a formal description of the `exact solution of
sliding problem windowsa the following additional
symbols are introduced:
I18
I1!35
M,6.
M1!35
maximal number of tasks in a problem
window (!)
set of tasks which are in a problem window
number of the currently "nally established stations (!)
number of stations which are needed in
the solution of a problem window (!)
M&*9
PartM
number of stations which are needed in
the solution of a problem window and
which are "nally established (!)
proportion of the maximal number of
"nally established stations to the number
of stations which are needed in the solution of a problem window (!).
An important assumption for the formal description which is given in Fig. 5 is the topological
numbering of the tasks, i.e. if i3F then h(i holds.
h
If the tasks are not numbered topologically then
they have to be renumbered "rst.
The small problems which have to be solved by
the use of an exact method are called `problem
windowsa. A problem window is de"ned by the "rst
IPW lowest numbered tasks of the set of the not yet
assigned tasks I3%45. If DI3%45D)IPW holds then the
current problem window is the last one. If DI3%45D
( IPW holds then the last problem window contains only DI3%45D tasks. As the tasks are numbered
topologically the precedence restrictions among the
set of tasks are not violated by this simple approach
for generating problem windows. From the
M1!35 stations generated in a solution of a problem
window the "rst M&*9 stations with their corresponding station loads are "nally established. The
number M&*9 is calculated by multiplying the
M1!35 stations by PartM. Because of the indivisibility of stations we have to round the result (see
step 3) (x
ay is the highest indivisible number
not higher than a). To avoid resolving of the
8
M. Amen / Int. J. Production Economics 68 (2000) 1}14
Fig. 5. Exact solution of sliding problem windows.
Fig. 6. Precedence graph for the example (cycle time c"10 TU/PU).
M. Amen / Int. J. Production Economics 68 (2000) 1}14
9
Fig. 7. Illustration of the exact solution of sliding problem windows.
Table 1
Optimal solution of the 1st problem window
Table 2
Optimal solution of the 2nd problem window
Station
m
Station
load
I4
m
Duration of
the operation
d4
m
(TU/PU)
Cost rate of
the station
k4
m
(MU/TU)
Station
m
Station
load
I4
m
Duration of
the operation
d4
m
(TU/PU)
Cost rate of
the station
k4
m
(MU/TU)
1
2
3
4
M1,
M3,
M5,
M8,
6
10
9
5
2
6
6
3
1
4
5
M5, 9, 13N
M8, 10, 11, 14N
M7, 12, 15N
10
10
9
6
3
6
2N
4, 6N
7, 9N
10N
M1!35 "
: 4 stations were needed to perform the tasks 1}10.
According to the formula M&*9 "
: maxM1, x
PartM ) M1!35
yN we
establish "nally M&*9 "
: maxM1, x0.7 ) 4y N"maxM1, x
2.8y N"2
stations. The tasks 1 and 2 are "nally assigned to station 1, the
tasks 3, 4 and 6 are "nally assigned to station 2. The assignments
of tasks to the stations 3 and 4 are abolished.
M1!35 "
: 3 stations were needed to perform the set of tasks in
this problem window I1!35. According to the formula
M&*9 "
: maxM1, x
PartM ) M1!35
yN
we
establish
"nally
M&*9 "
: maxM1, x
0.7 ) 3y N"maxM1, x
2.1y N"2 stations. The
tasks 5, 9 and 13 are "nally assigned to station 3, the tasks 8, 10,
11 and 14 are "nally assigned to station 4. The assignments of
tasks to the station 5 are abolished.
10
M. Amen / Int. J. Production Economics 68 (2000) 1}14
5.2. An example of the solution process
Table 3
Optimal solution of the 3rd problem window
Station
m
Station
load
I4
m
Duration of
the operation
d4
m
(TU/PU)
Cost rate of
the station
k4
m
(MU/TU)
5
6
7
M7, 15, 16N
M18, 19N
M12, 17, 20N
10
6
8
6
3
6
M1!35 "
: 3 stations were needed to perform the set of tasks in
this last problem window I1!35. Because this is the last problem
window we establish "nally all M1!35 stations. The tasks 7, 15
and 16 are "nally assigned to station 5, the tasks 18 and 19 are
"nally assigned to station 6, the tasks 12, 17 and 20 are "nally
assigned to station 7.
same problem window, which otherwise may occur
according to the parameter constellations, we
have to ensure that at least one station of
the solution is established "nally. Therefore
we set M&*9 "
: maxM1, x
PartM ) M1!35yN. The tasks
assignments of the M1!35}M&*9 last stations of
a solution of a problem window are abolished. In
the next problem window a balance for the
IPW lowest numbered not yet assigned tasks has to
be generated and the "rst M&*9 stations have to be
"nally established. This procedure continues until
all I tasks are "nally assigned to a station.
The method described in Section 5.1 will now be
illustrated by an example. Fig. 6 shows the precedence graph of the problem instance together with
the durations and the cost rates of the tasks. A precedence graph G"(M1,2, IN, R) consists of the set
of tasks M1,2, IN and the set of precedence relations R. A precedence relation with i as technological immediate follower of h, i3F , is illustrated by
h
an arrow (h, i) which is directed from h to i [21,
p. 685]. The cycle time for this instance is given by
c"10 TU/PU. Note that the tasks of the instance
are numbered topologically.
The 20-task-instance will be solved by a version
of the method which works with a maximum of
IPW"10 tasks in each problem window. From
a solution of a problem window about 70% of the
station loads generated were "nally established, i.e.
PartM"0.7.
In our implementation of this method we used
the exact backtracking method suggested by Amen
[5,6] * this is the only existent method to generate
an optimal solution for a cost-oriented assembly
line balancing instance. The initial upper bound for
the minimum cost per product unit which is needed
in this exact method was obtained from the heuristic
solution of the problem window by the use of the new
priority rule `best change of idle costa (P-MinKI).
Table 4
Final total solution
Station m
Station load I4
m
Duration of the operation d4
m
(TU/PU)
Cost rate of the station k4
m
(MU/TU)
From solution of problem
window no.
1
2
M1, 2N
M3, 4, 6N
6
10
2
6
1
1
3
4
M5, 9, 13N
M8, 10, 11, 14N
10
10
6
3
2
2
5
6
7
M7, 15, 16N
M18, 19N
M12, 17, 20N
10
6
8
***
60
6
3
6
***
32
3
3
3
***
k
M
10 ) 32"
320
7
MU/PU
Sum
Cost
Stations
M. Amen / Int. J. Production Economics 68 (2000) 1}14
11
Fig. 8. Phase 1 of wage rate smoothing.
Fig. 7 gives a comprehensive illustration of the
solution process for this example. As the exact
method is beyond the topic of this paper the
enumeration process and the dominance criteria
of this backtracking-procedure are not explained
here. In fact even for the solution of the "rst problem window with 10 task approximately 100 iterations were needed. Because to give detailed
information on the exact method is much spaceand time-consuming, it is left for further reading
(see [5,6]).
(1) Dexnition and solution of the xrst problem
window. The "rst problem window consists of the
"rst IPA"10 lowest numbered (not yet "nally
assigned) tasks. These are the tasks 1}10. Using
the exact method we get the solution shown in
Table 1.
(2) Dexnition and solution of the second problem
window. First we have to calculate the set of not yet
"nally assigned tasks I3%45. These are the tasks 5 and
7}20. The "rst IPA"10 lowest numbered task
among the set I3%45 are taken into the set of tasks of
this problem window I1!35. Therefore, we get
I1!35 "
: M5, 7, 8, 9, 10, 11, 12, 13, 14, 15N. Again, we
solve this problem window with the exact method
and calculate the solution shown in Table 2.
(3) Dexnition and solution of the third problem
window. First we have to calculate the set of not
yet "nally assigned tasks I3%45. These are the tasks 7,
12 and 15}20. As the number of not "nally assigned
task DI3%45D "8 is not higher than IPA"10, the
third problem window is the last one. Therefore, we
get I1!35 "
: I3%45"M7, 12, 15, 16, 17, 18, 19, 20N.
Again, we solve this problem window with the
12
M. Amen / Int. J. Production Economics 68 (2000) 1}14
Fig. 9. Phase 2 of wage rate smoothing.
exact method and obtain the solution shown in
Table 3.
(4) Final total solution. Combining the "nal assignments which are calculated in the solutions of the
problem windows we get the "nal total solution
(Table 4).
As the minimum cost is 310 MU/PU, the solution generated by this method has the second
lowest possible cost (320 MU/PU). To give additional information we have found that there exist
18 optimal solutions for this problem instance. In
each of the optimal solutions 7 stations were
needed. The lowest realizable number of stations
is 6 (94 solutions). With 6 stations a minimum of
330 MU/PU for the cost is calculated. Note that
the stations 1 and 6 are not loaded maximally. This
is another evidence that loading the stations maxi-
mally could prevent one from generating low-cost
solutions [5, p. 225; 6].
6. Summary and outlook
This paper is concerned with the existent and
new heuristic methods for solving the cost-oriented
assembly line balancing problem. The methods are
classi"ed and described in detail. Two new heuristic
methods are presented: The priority rule `best
change of idle costa and the more sophisticated
method `exact solution of sliding problem windowsa.
The sliding problem window technique is a new
class of its own. It can be taken as a metaheuristic
because its possible application is not only for the
assembly line balancing, but also for some other
problems with precedence relations. In a further
M. Amen / Int. J. Production Economics 68 (2000) 1}14
13
Fig. 10. Phase 3 of wage rate smoothing.
paper a comparison of the heuristic methods according to the solution quality and the computing
time is presented [22].
Appendix
The formal discriptions of the phases 1}3 are
given in Figs. 8}10.
References
[1] S. Ghosh, R.J. Gagnon, A comprehensive literature review
and analysis of the design, balancing and scheduling of
assembly systems, International Journal of Operations
Research 27 (1989) 637}670.
[2] I. Baybars, A survey of exact algorithms for the simple
assembly line balancing problem, Management Science 32
(1986) 909}932.
[3] R. Ste!en, Produktionsplanung bei Flie{bandfertigung,
Gabler, Wiesbaden, 1977.
14
M. Amen / Int. J. Production Economics 68 (2000) 1}14
[4] O. Rosenberg, H. Ziegler, A comparison of heuristic algorithms for cost-oriented assembly line balancing, Zeitschrift fuK r Operations Research 36 (1992) 477}495.
[5] M. Amen, Ein exaktes Verfahren zur kostenorientierten
Flie{bandabstimmung, in: U. Zimmermann et al., (Eds.),
Operations Research Proceedings 1996, Springer, Berlin,
1997, pp. 224}229.
[6] M. Amen, An exact method for cost-oriented assembly line
balancing, International Journal of Production Economics
64 (2000) 187}195.
[7] T.S. Wee, M.J. Magazine, Assembly line balancing as generalized bin packing, Operations Research Letters 1 (1982)
56}58.
[8] St.T. Hackman, H.J. Magazine, T.S. Wee, Fast, e!ective
algorithms for simple assembly line balancing problems,
Operations Research 37 (1989) 916}924.
[9] A. Scholl, Balancing and Sequencing of Assembly Lines,
2nd ed., Physica-Verlag, Heidelberg, 1999.
[10] J.R. Jackson, A computing procedure for a line balancing
problem, Management Science 2 (1956) 261}271.
[11] F.N. Silverman, The e!ects of stochastic work times on the
assembly line balancing problem, Ph.D. Thesis, Columbia
University, New York, 1974.
[12] F.M. Tonge, Assembly line balancing using probabilistic
combinations of heuristics, Management Science 11 (1965)
727}735.
[13] W.B. Helgeson, D.P. Birnie, Assembly line balancing using
the ranked positional weight technique, Journal of Industrial Engineering 12 (1961) 394}398.
[14] R. Hahn, Produktionsplanung bei Linienfertigung, Walter
de Gruyter, Berlin, 1972.
[15] Heizmann, J., Soziotechnologische Ablaufplanung verketteter
Fertigungsnester zur ErhoK hung der FlexibilitaK t von Montage-Flie{linien, Jochem Heizmann Verlag, Karlsruhe, 1981.
[16] E.G. Co!man et al., Performance bounds for leveloriented two-dimensional packing algorithms, SIAM
Journal on Computing 9 (1980) 808}826.
[17] C. Imboden, A. Leibundgut, P. Siegenthaler, Klassi"kation heuristischer Prinzipien: Ein methodologischer Beitrag zur Entwicklung von heuristischen Verfahren, Die
Unternehmung 32 (1978) 295}330.
[18] L.B.J.M. Sturm, An improved method for balancing assembly lines, Working paper R 70/1, Interfaculty for Graduate Studies in Management, Rotterdam, 1970.
[19] W.H. Thomas, N.R. Reeve, Balancing continuous stochastic assembly lines, in: American Institute of Industrial
Engineers, Technical Papers, 23rd Annual Conference and
Convention of the American Institute of Industrial Engineers, Anaheim, California, 1972, pp. 409}419.
[20] N.R. Reeve, W.H. Thomas, Balancing stochastic assembly
lines, AIIE [American Institute of Industrial Engineers]
Transactions 5 (1973) 223}229.
[21] R. Reiter, On assembly-line balancing problems, Operations Research 17 (1969) 685}700.
[22] M. Amen, Heuristic methods for cost-oriented assembly
line balancing: A comparison on solution quality and
computing time, International Journal of Production
Economics 69 (2001) Forthcoming.
Heuristic methods for cost-oriented assembly
line balancing: A survey
Matthias Amen*
Institut fu( r Unternehmensrechnung und Controlling, University of Berne, Engehaldenstrasse 4, CH-3012 Bern, Switzerland
Received 17 March 1998; accepted 21 September 1998
Abstract
This paper is concerned with cost-oriented assembly line balancing. This problem occurs especially in the "nal
assembly of automotives, consumer durables or personal computers, where production is still very labour-intensive, and
where the wage rates depend on the requirements and quali"cations to ful"l the work. First a short problem description is
presented. After that a classi"cation of existent and new heuristic methods for solving this problem is given. The heuristic
methods presented in this paper are described in detail. A new priority rule called `best change of idle costa is proposed.
This priority rule di!ers from the existent priority rules because it is the only one which considers that production cost
are the result of both, production time and cost rates. Furthermore a new sophisticated method called `exact solution of
sliding problem windowsa is presented. The solution process is illustrated by an example, showing how this metaheuristic
works together with an exact method. ( 2000 Elsevier Science B.V. All rights reserved.
Keywords: Assembly line balancing; Cost-oriented production planning; Heuristic methods
1. Introduction
In an assembly line the product units move with
a constant transportation speed through the consecutive stations. The total work content to be
performed by the production system has been split
up into economical indivisible work elements
which are called tasks. Among the set of tasks there
exist technological precedence relations. The set of
tasks to be performed in the same station is called
an operation or a station load. The time to perform
an operation is restricted by the cycle time. The
* Tel.: #41-31-631-85-01; fax: #41-31-631-37-80.
E-mail address: matthias.amen@iuc.unibe.ch (M. Amen)
assembly line balancing problem consists in allocating the tasks to the stations subject to the
technological precedence relations, the cycle time
restriction of the stations and the indivisibility of
the tasks.
In literature the objective usually is to minimize
the number of stations in a line for a "xed cycle
time [1, p. 650]. In other words: The objective is to
minimize the total idle time of the total capacity
provided by the sum of the stations of the line [2,
p. 911]. Therefore, this is called time-oriented assembly line balancing. The objective in cost-oriented
assembly line balancing which is considered in this
paper is to minimize the total cost per product unit
[3}6].
In general, assembly is a labour-intensive kind of
production. Therefore, in cost-oriented assembly
0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 9 9 ) 0 0 0 9 5 - X
2
M. Amen / Int. J. Production Economics 68 (2000) 1}14
line balancing, the labour cost has to be analyzed in
detail: The wage rate of each operation depends on
the maximal point value of the tasks assigned to
a station. As the tasks di!er in their di$culty,
there are di!erences in the point values and
thus in the corresponding wage rates. The worker
has to be paid for the whole cycle time irrespective
of the duration of the operation. So the total
labour cost per product unit is the sum of the
wage rates of all stations multiplied by the cycle
time. Further it is assumed that the cost of
capital depends on the overall line length and that
all stations have the same length. All other costs
(e.g. material cost) can be assumed to be independent of the division of labour and of the line length
[3, p. 82]. Introducing the following symbols:
I
number of tasks (!)
i
index for the tasks, i"1(1)I
M number of stations (!)
m
index for the stations, m"1(1)M
set of tasks, assigned to station m, m"1(1)M
I4
m
c
cycle time (TU/PU)
k
total cost per unit (MU/PU)
cost rate of station m, m"1(1)M (MU/TU)
k4
m
k43 cost of capital per station (MU/PU)
k48 wage rate of station m, m"1(1)M (MU/TU)
m
cost rate of task i, i"1(1)I (MU/TU)
k5
i
k58 wage rate of task i, i"1(1)I (MU/TU)
i
(Note: Dimensions: TU, time units; PU, product
units; MU, monetary units; Labels: r, cost of capital; s, station; t, task; w, wage rate.)
The total cost per product unit can be calculated as
A
B
M
k" + ck48 #Mk43
m
m/1
with k48"maxMk58Di3Is N.
m
i
m
Since the conveyor speed is "xed by the cycle
time, the cost of capital for a single station can
be transformed into a cost rate per time unit.
The total cost of capital can be formulated as
Mk43"cM(k43/c). Thus for the purpose of capacity
balancing the relevant cost rates per time unit are
made up by the sum of the capital cost per time unit
and the task-depended wage rates: k5"k58#k4#/c
i
i
and k4 "k48#k4#/c. This simpli"es the calculation
m
m
of the total cost per product unit to
M
M
k" + ck4 " + c maxMk5Di3I4 N.
m
i
m
m/1
m/1
The total cost per product unit k"+M ck4
m/1 m
di!er from the term +I d5k5 only by the idle cost
i/1 i i
per product unit which are caused by idle time
and/or cost rate di!erences of the tasks assigned to
the same station. Therefore, the objective to minimize the total cost per product unit is equivalent to
minimize the total idle cost per product unit.
2. Classi5cation of heuristic methods for solving the
cost-oriented assembly line balancing problem
As the NP-complete bin-packing problem can be
reduced to the time-oriented assembly line balancing problem in polynomial time, the time-oriented
assembly line balancing problem is NP-hard [7,
p. 56]. This also holds for the cost-oriented problem [4, p. 479]. Thus it is justi"ed to develop
heuristic solution methods.
There exist two di!erent general approaches for
assembly line balancing. In the station-oriented approach in each step of the solution process only one
station is considered. Therefore we have to look for
tasks which have to be assigned to the current
station. In the task-oriented approach in each step of
the solution process a selected task is considered.
Therefore we have to look for the station to which
the current task has to be assigned to [8, pp.
917}918; 9, pp. 182}184]. All of the methods described in this paper use the station-oriented approach. Fig. 1 gives an overview of the existent and
new methods which can be classi"ed as follows:
Most of the methods can be characterized as
random choice/priority rule methods. In each step of
the solution process all of the methods of this
category but one choose one of alternative tasks for
assignment. For determining the task either random choice or priority rules are used. The priority
rule methods can be distinguished into methods
that make use of one problem-oriented priority
rule and methods that use several problem-oriented
priority rules. Priority rules are called problemoriented if they consider structural aspects of the
M. Amen / Int. J. Production Economics 68 (2000) 1}14
3
Fig. 1. Heuristic methods for cost-oriented assembly line balancing.
problem. The set of problem-oriented priority rules
consists of three subclasses. The time-oriented priority rules consider only the durations and/or the
precedence relations of the tasks, the cost rateoriented priority rules consider only the cost rates of
the tasks. Due to the fact that cost are the product
of both, cost-rate and duration, the new costoriented priority rule **best change of idle cost++
which will be presented in the next section takes
both aspects into account. In methods with several
problem-oriented priority rules the single rules are
used in lexicographic order, i.e. problem-oriented
priority rules are used as tie-breakers. All priority
rule methods use a simple non-problem-oriented
priority rule as a "nal tie-breaker.
One priority method di!ers from the other as in
the major steps of the solution process one of several alternative station loads has to be chosen by
the use of di!erent priority rules which consider the
situation in the alternative station loads. The alternative station loads are generated by the use of
several problem-oriented priority rules to choose
one of alternative tasks. Thus this method is a
further development of the previous mentioned
methods.
The new method called **exact solution of sliding
problem windows++ is quite di!erent from the heuristic methods mentioned before because it is a heuristic version of an exact method. With respect to the
similarities of the heuristic methods, the methods
are grouped into the following classes:
f Z methods with random choice task assignment;
f P methods with one problem-oriented priority
rule;
f H methods with several problem-oriented priority rules;
f F methods with exact solution of sliding problem
windows;
f E exact method.
Class H contains both, the simple methods which
choose one of alternative tasks and the more advanced method which makes use of these simple
methods when generating alternative station loads.
In the next section the methods of classes Z and
P and the simple methods of class H are presented.
Section 4 is concerned with the more advanced
method of class H. Section 5 deals with the methods
of class F. Class E consists of only one method. As
exact methods are beyond the topic of this paper
4
M. Amen / Int. J. Production Economics 68 (2000) 1}14
assignable tasks is empty, the current station
is **maximally loaded++ [10, p. 266; 8, p. 918].
Then a new station is established and the set of
assignable tasks has to be updated again. Each
method stops after all tasks have been assigned to a
station.
The random choice method (class Z) [11,
pp. 25}35 and 44}46] can be used several times for
solving the same problem instance. Among the set
of randomly generated solutions for the same problem instance only the best one has to be considered
for realization.
For a formal description of the priority rule
methods, the following additional symbols are introduced:
Fig. 2. Procedure for assigning tasks to stations.
this method is not presented in this paper (for
further reading see [5,6]. The exact method works
as a subalgorithm for the methods of class F.
I!44*'/ set of assignable tasks
duration of task i, i"1(1)I (TU/PU)
d5
i
F
set of all technological immediate following
i
tasks of task i, i"1(1)I
*k
change of idle cost if task i will be assigned
i
to the currently considered station, i3I!44*'/
(MU/PU)
r
ranked positional weight of task i, i"1(1)I
i
(TU/PU)
With this symbols the following problem-oriented
priority rules of class P can be de"ned:
z P-MaxD
3. Methods with choice of one of alternative tasks
All methods with choice of one of alternative
tasks in each step of the solution process follow the
same procedure when assigning tasks to stations.
The procedure is shown in Fig. 2.
As the station loads are generated successively, in
each step of the procedure only one station is considered (station-oriented approach). The tasks which
have no predecessor task or only predecessor tasks
already assigned to the current or to an earlier
station are called available. The available tasks
which have a duration not longer than the remaining idle time of the considered station are called
assignable. Among the set of assignable tasks one
task is chosen according to the method used. After
assigning this task the sets of available and assignable tasks are updated immediately. If the set of
z P-MaxR
Maximal task duration [12, p. 728],
i.e. maxMd5Di3I!44*'/N.
i
Maximal ranked positional weight
([13, p. 395 in the modi"ed version
of Hahn], [14, pp. 44}46]), i.e.
maxMr Di3I!44*'/N, where
i
G
d5#+ i r ,
j|F j
r" i
i
d5,
i
z P-MaxF
if F OH,
i
if F "H.
i
Maximal number of immediate
followers [12, p. 728], i.e.
max MDF D D i3I!44*'/N.
i
z P-MaxKt Maximal cost rate [4, p. 481], i.e.
maxMk5Di3I!44*'/N.
i
z P-MinKt Minimal cost rate [3, pp. 106}107],
i.e. min Mk5Di3I!44*'/N.
i
z P-MinKts Minimal absolute di!erence to the
current station cost rate [3, pp.
106}107], i.e. minMDk5!k4 D D i3I!44*'/N.
i
m
M. Amen / Int. J. Production Economics 68 (2000) 1}14
5
both, the cost rates and the time needed by the
workers of the line.
Within class H there exist three priority rule
methods with the choice of one of alternative tasks
in each step of the procedure. The lexicographic
orders of priority rules of these heuristic methods
are given below. For the heuristic methods of Steffen and Heizmann the lexicographic order di!ers
according to the fact whether the station is just
established (empty station) or the station has been
established at an earlier step in the solution process
(partially "lled station).
z H-Stef
Fig. 3. Calculation of *k .
i
z P-MinKI
Best change of idle cost, i.e.
minM*k Di3I!44*'/N, where
i
if k5)k4 ,
!d5k5 ,
m
i
i i
*k "
i
(k5!k4 )c!d5 k5 , if k5'k4 .
m
i
i i
m
i
The non-problem-oriented priority rule
G
z P-MinI
Minimal task number [12, p. 729],
i.e. min MiDi3I!44*'/N (Usually the
tasks are numbered topologically according to the precedence relations.)
is used as "nal tie breaker in all priority rule
methods. While applying the new priority rule `best
change of idle costa (P-MinKI) both decreases and
increases can occur, which has to be accepted. Not
choosing an assignable task with *k '0 and esi
tablishing a new station in some cases may cause an
extra station and therefore higher overall idle cost.
The calculation of *k is illustrated in Fig. 3.
i
P-MinKts and P-MinKI are dynamic rules because they depend on the current cost rate of the
station. For these rules it is necessary to calculate
not only the remaining idle time of the current
station after each assignment, but also to update
the cost rate of the station immediately. While
P-MaxD, P-MaxR and P-MaxF are only timeoriented and P-MaxKt, P-MinKt and P-MinKts
depend only on the cost rates, P-MinKI is the only
priority rule which considers that cost results from
Heuristic method of Ste!en [3, pp.
105}110]:
Lexicographic
Lexicographic order
order of priority of priority rules in
rules in case of case of already
just established established stations
stations (station (station is partially
is empty):
"lled with tasks):
1. P-MaxR
1. P-MinKts
2. P-MinI
2. P-MinKt
3. P-MaxR
4. P-MinI
z H-Heiz Heuristic method of Heizmann [15,
pp. 110}124]:
Lexicographic order
Lexicographic
order of priority of priority rules in
rules in case of case of already
just established established stations
stations (station (station is partially
"lled with tasks):
is empty):
1. P-MaxD
1. Identity of task
cost rate and current
station cost rate (i.e.
Dk5!k4 D"0 is
m
i
required).
2. P-MinI
2. P-MaxD
3. P-MinI
If there is no task which has the same
cost rate as the current station cost rate
in the case of an already established
station (station is partially "lled with
tasks), then the priority rule P-MaxD is
used together with the tie-breaking rule
P-MinI.
6
z H-WR
M. Amen / Int. J. Production Economics 68 (2000) 1}14
Heuristic method `wage ratea of Rosenberg/Ziegler [4, p. 481]:
1. P-MaxKt
2. P-MaxD
3. P-MinI
This heuristic method is a modi"cation
of the well-known `"rst-"t decreasing
heighta-method suggested for solving
two-dimensional packing problems [16,
p. 810]. The method is called `wage
ratea because Rosenberg/Ziegler [4]
consider only labour cost.
I!44*'/,105
m,w,u
d4,105
m,w,u
k4,105
m,w,u
"xed
4. A method with choice of one of alternative
station loads
A priority-rule method with choice of one of
alternative station loads is the `wage rate smoothinga method (H-WRS) [4, pp. 481}484]. The
method consists of a three-phase algorithm. In
order to give a formal description of this method
the following symbols are introduced:
I3%45
set of not yet "nally assigned tasks
I4,105,1)!4%2 set of tasks which de"nes the potential
m
station load of station m in phase 2,
m"1(1)M
I4,105,1)!4%3 set of tasks which de"nes the potential
m
station load of station m in phase 3,
m"1(1)M
=
number of di!erent cost rates (!)
w
index of the cost rate, w3M1,2, =N
set of assignable tasks with the cost
I!44*'/,l
w
rate kl , w"1(1)=
w
;
number of potential station loads for
w
the currently considered task cost rate
kl (!), w"1(1)=
w
u
index of the potential station loads,
u3M1,2, ; N
w
set of tasks which de"nes the uth poI4,,,105
m,w,u
tential station load for the considered
cost rate kl of the station m in the
w
phases 1 and 3, u"1(1); , w"1(1)=,
w
m"1(1)M
set of tasks which de"nes the uth potenI4,,,105,2
m,w,u
tial station load for the considered cost
rate kl of the station m in the phase 2,
w
u"1(1); , w"1(1)=, m"1(1)M
w
set of tasks with task cost rate kl that
w
are assignable to station m if
the current station load is de"ned
by the set of tasks I4,,,105, u"1(1); ,
m,w,u
w
w"1(1)=, m"1(1)M
work content of the current station
load I4,,,105 (TU/PU), u"1(1); ,
m,w,u
w
w"1(1)=, m"1(1)M
cost rate of the potential station
load I4,,,105 (MU/TU), u"1(1); ,
m,w,u
w
w"1(1)=, m"1(1)M
variable which indicates in phase 1
whether phase 1 has to be restarted or
whether phase 2 has to be started (!)
Fig. 4 shows the major steps of the 3-phase
algorithm. The formal descriptions of the phases
1}3 are given in the appendix.
In phase 1 of the procedure, potential station
loads are established. In phases 2 and 3 the potential station loads which have been calculated in
phase 1 are taken and further tasks are assigned. In
phase 1 each potential station load consists only of
tasks which have identical cost rates. The phases 2
and 3 are completely separated from each other. In
phase 2 a potential station load taken from phase 1
is "lled further only with tasks which have the same
cost rate as the station cost rate. In phase 3 a potential station load taken from phase 1 is "lled further
with tasks which could have cost rates di!erent
from the station cost rate of phase 1. In step 5 a station load has to be chosen from the two alternative
station loads which have been obtained from the
phases 2 and 3. After a potential station load is
"nally "xed, in step 5 the algorithm moves back to
phase 1. Within phase 1 stopping-criteria are to be
checked at two di!erent steps.
5. A method with exact solution of sliding
problem windows
A new heuristic method which is quite di!erent
from those described in the foregoing sections is the
`exact solution of sliding problem windowsa (class
F). This heuristic is an application of the `working
M. Amen / Int. J. Production Economics 68 (2000) 1}14
7
Fig. 4. Wage rate smoothing.
forward techniquea [17, pp. 315}316]. The key concept is to solve consecutive small problems by the
use of an exact method. A part of the optimal
solution of a small problem will be "xed "nally.
Then the next small problem will be de"ned and
solved optimally. This process continues until the
last small problem is solved. Similar methods for
solving di!erent assembly line balancing problems
were suggested in [18}20].
The idea to develop a heuristic application of an
exact method derives from the fact that all existent
heuristic methods load the stations maximally, but
that the cost-oriented optimal solution could be missed if the stations are "lled in this way [5, p. 225; 6,
1998a]. The exact method used is the only existent
method in which the stations are not necessarily
loaded maximally.
5.1. Formal description of the method
For a formal description of the `exact solution of
sliding problem windowsa the following additional
symbols are introduced:
I18
I1!35
M,6.
M1!35
maximal number of tasks in a problem
window (!)
set of tasks which are in a problem window
number of the currently "nally established stations (!)
number of stations which are needed in
the solution of a problem window (!)
M&*9
PartM
number of stations which are needed in
the solution of a problem window and
which are "nally established (!)
proportion of the maximal number of
"nally established stations to the number
of stations which are needed in the solution of a problem window (!).
An important assumption for the formal description which is given in Fig. 5 is the topological
numbering of the tasks, i.e. if i3F then h(i holds.
h
If the tasks are not numbered topologically then
they have to be renumbered "rst.
The small problems which have to be solved by
the use of an exact method are called `problem
windowsa. A problem window is de"ned by the "rst
IPW lowest numbered tasks of the set of the not yet
assigned tasks I3%45. If DI3%45D)IPW holds then the
current problem window is the last one. If DI3%45D
( IPW holds then the last problem window contains only DI3%45D tasks. As the tasks are numbered
topologically the precedence restrictions among the
set of tasks are not violated by this simple approach
for generating problem windows. From the
M1!35 stations generated in a solution of a problem
window the "rst M&*9 stations with their corresponding station loads are "nally established. The
number M&*9 is calculated by multiplying the
M1!35 stations by PartM. Because of the indivisibility of stations we have to round the result (see
step 3) (x
ay is the highest indivisible number
not higher than a). To avoid resolving of the
8
M. Amen / Int. J. Production Economics 68 (2000) 1}14
Fig. 5. Exact solution of sliding problem windows.
Fig. 6. Precedence graph for the example (cycle time c"10 TU/PU).
M. Amen / Int. J. Production Economics 68 (2000) 1}14
9
Fig. 7. Illustration of the exact solution of sliding problem windows.
Table 1
Optimal solution of the 1st problem window
Table 2
Optimal solution of the 2nd problem window
Station
m
Station
load
I4
m
Duration of
the operation
d4
m
(TU/PU)
Cost rate of
the station
k4
m
(MU/TU)
Station
m
Station
load
I4
m
Duration of
the operation
d4
m
(TU/PU)
Cost rate of
the station
k4
m
(MU/TU)
1
2
3
4
M1,
M3,
M5,
M8,
6
10
9
5
2
6
6
3
1
4
5
M5, 9, 13N
M8, 10, 11, 14N
M7, 12, 15N
10
10
9
6
3
6
2N
4, 6N
7, 9N
10N
M1!35 "
: 4 stations were needed to perform the tasks 1}10.
According to the formula M&*9 "
: maxM1, x
PartM ) M1!35
yN we
establish "nally M&*9 "
: maxM1, x0.7 ) 4y N"maxM1, x
2.8y N"2
stations. The tasks 1 and 2 are "nally assigned to station 1, the
tasks 3, 4 and 6 are "nally assigned to station 2. The assignments
of tasks to the stations 3 and 4 are abolished.
M1!35 "
: 3 stations were needed to perform the set of tasks in
this problem window I1!35. According to the formula
M&*9 "
: maxM1, x
PartM ) M1!35
yN
we
establish
"nally
M&*9 "
: maxM1, x
0.7 ) 3y N"maxM1, x
2.1y N"2 stations. The
tasks 5, 9 and 13 are "nally assigned to station 3, the tasks 8, 10,
11 and 14 are "nally assigned to station 4. The assignments of
tasks to the station 5 are abolished.
10
M. Amen / Int. J. Production Economics 68 (2000) 1}14
5.2. An example of the solution process
Table 3
Optimal solution of the 3rd problem window
Station
m
Station
load
I4
m
Duration of
the operation
d4
m
(TU/PU)
Cost rate of
the station
k4
m
(MU/TU)
5
6
7
M7, 15, 16N
M18, 19N
M12, 17, 20N
10
6
8
6
3
6
M1!35 "
: 3 stations were needed to perform the set of tasks in
this last problem window I1!35. Because this is the last problem
window we establish "nally all M1!35 stations. The tasks 7, 15
and 16 are "nally assigned to station 5, the tasks 18 and 19 are
"nally assigned to station 6, the tasks 12, 17 and 20 are "nally
assigned to station 7.
same problem window, which otherwise may occur
according to the parameter constellations, we
have to ensure that at least one station of
the solution is established "nally. Therefore
we set M&*9 "
: maxM1, x
PartM ) M1!35yN. The tasks
assignments of the M1!35}M&*9 last stations of
a solution of a problem window are abolished. In
the next problem window a balance for the
IPW lowest numbered not yet assigned tasks has to
be generated and the "rst M&*9 stations have to be
"nally established. This procedure continues until
all I tasks are "nally assigned to a station.
The method described in Section 5.1 will now be
illustrated by an example. Fig. 6 shows the precedence graph of the problem instance together with
the durations and the cost rates of the tasks. A precedence graph G"(M1,2, IN, R) consists of the set
of tasks M1,2, IN and the set of precedence relations R. A precedence relation with i as technological immediate follower of h, i3F , is illustrated by
h
an arrow (h, i) which is directed from h to i [21,
p. 685]. The cycle time for this instance is given by
c"10 TU/PU. Note that the tasks of the instance
are numbered topologically.
The 20-task-instance will be solved by a version
of the method which works with a maximum of
IPW"10 tasks in each problem window. From
a solution of a problem window about 70% of the
station loads generated were "nally established, i.e.
PartM"0.7.
In our implementation of this method we used
the exact backtracking method suggested by Amen
[5,6] * this is the only existent method to generate
an optimal solution for a cost-oriented assembly
line balancing instance. The initial upper bound for
the minimum cost per product unit which is needed
in this exact method was obtained from the heuristic
solution of the problem window by the use of the new
priority rule `best change of idle costa (P-MinKI).
Table 4
Final total solution
Station m
Station load I4
m
Duration of the operation d4
m
(TU/PU)
Cost rate of the station k4
m
(MU/TU)
From solution of problem
window no.
1
2
M1, 2N
M3, 4, 6N
6
10
2
6
1
1
3
4
M5, 9, 13N
M8, 10, 11, 14N
10
10
6
3
2
2
5
6
7
M7, 15, 16N
M18, 19N
M12, 17, 20N
10
6
8
***
60
6
3
6
***
32
3
3
3
***
k
M
10 ) 32"
320
7
MU/PU
Sum
Cost
Stations
M. Amen / Int. J. Production Economics 68 (2000) 1}14
11
Fig. 8. Phase 1 of wage rate smoothing.
Fig. 7 gives a comprehensive illustration of the
solution process for this example. As the exact
method is beyond the topic of this paper the
enumeration process and the dominance criteria
of this backtracking-procedure are not explained
here. In fact even for the solution of the "rst problem window with 10 task approximately 100 iterations were needed. Because to give detailed
information on the exact method is much spaceand time-consuming, it is left for further reading
(see [5,6]).
(1) Dexnition and solution of the xrst problem
window. The "rst problem window consists of the
"rst IPA"10 lowest numbered (not yet "nally
assigned) tasks. These are the tasks 1}10. Using
the exact method we get the solution shown in
Table 1.
(2) Dexnition and solution of the second problem
window. First we have to calculate the set of not yet
"nally assigned tasks I3%45. These are the tasks 5 and
7}20. The "rst IPA"10 lowest numbered task
among the set I3%45 are taken into the set of tasks of
this problem window I1!35. Therefore, we get
I1!35 "
: M5, 7, 8, 9, 10, 11, 12, 13, 14, 15N. Again, we
solve this problem window with the exact method
and calculate the solution shown in Table 2.
(3) Dexnition and solution of the third problem
window. First we have to calculate the set of not
yet "nally assigned tasks I3%45. These are the tasks 7,
12 and 15}20. As the number of not "nally assigned
task DI3%45D "8 is not higher than IPA"10, the
third problem window is the last one. Therefore, we
get I1!35 "
: I3%45"M7, 12, 15, 16, 17, 18, 19, 20N.
Again, we solve this problem window with the
12
M. Amen / Int. J. Production Economics 68 (2000) 1}14
Fig. 9. Phase 2 of wage rate smoothing.
exact method and obtain the solution shown in
Table 3.
(4) Final total solution. Combining the "nal assignments which are calculated in the solutions of the
problem windows we get the "nal total solution
(Table 4).
As the minimum cost is 310 MU/PU, the solution generated by this method has the second
lowest possible cost (320 MU/PU). To give additional information we have found that there exist
18 optimal solutions for this problem instance. In
each of the optimal solutions 7 stations were
needed. The lowest realizable number of stations
is 6 (94 solutions). With 6 stations a minimum of
330 MU/PU for the cost is calculated. Note that
the stations 1 and 6 are not loaded maximally. This
is another evidence that loading the stations maxi-
mally could prevent one from generating low-cost
solutions [5, p. 225; 6].
6. Summary and outlook
This paper is concerned with the existent and
new heuristic methods for solving the cost-oriented
assembly line balancing problem. The methods are
classi"ed and described in detail. Two new heuristic
methods are presented: The priority rule `best
change of idle costa and the more sophisticated
method `exact solution of sliding problem windowsa.
The sliding problem window technique is a new
class of its own. It can be taken as a metaheuristic
because its possible application is not only for the
assembly line balancing, but also for some other
problems with precedence relations. In a further
M. Amen / Int. J. Production Economics 68 (2000) 1}14
13
Fig. 10. Phase 3 of wage rate smoothing.
paper a comparison of the heuristic methods according to the solution quality and the computing
time is presented [22].
Appendix
The formal discriptions of the phases 1}3 are
given in Figs. 8}10.
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